Abstract

We consider the online (over time) scheduling of equal length jobs on a bounded parallel batch machine with batch capacity to minimize the time by which all jobs have been delivered with limited restart. Here, “restart” means that a running batch may be interrupted, losing all the work done on it, and jobs in the interrupted batch are then released and become independently unscheduled jobs, called restarted jobs. “Limited restart” means that a running batch which contains some restarted jobs cannot be restarted again. When , we propose a best possible online algorithm with a competitive ratio of , where is the positive solution of . When , we present a best possible online algorithm with a competitive ratio of , where is the positive solution of .

1. Introduction

In this paper, we consider the online scheduling of equal length jobs on a bounded parallel batch machine with batch capacity to minimize the time by which all jobs have been delivered with limited restart. Here, a bounded parallel batch machine is a machine which can process up to jobs simultaneously as a batch, where is the batch capacity. The processing time of a batch is the longest processing time of jobs in the batch. Jobs in a batch have the same starting time and the same completion time. According to the characteristic of batch capacity , there are the bounded model and the unbounded model. The bounded model means that is finite; that is, . The unbounded model means that can be as large as possible; that is, . In this paper, we study the bounded model, that is, bounded parallel batch machine.

In this paper, online means that each job becomes available at its arrival time which is unknown beforehand and the characteristics of the job are unknown until it arrives. Each job has an arrival time , a processing time , and a delivery time . In our research, we assume that all jobs have equal processing times. By scaling, we may assume that each job has a processing time . Each job needs to be processed on the bounded parallel batch machine and then delivered to the destination by a delivery vehicle. Here we assume that there are sufficiently many delivery vehicles, which means that a job can be transported as soon as it is completed on the bounded parallel batch machine. Let be the completion time of on the bounded parallel batch machine and let be the time by which has been delivered. Then . Let . Our objective is to minimize , that is, the time by which all jobs have been delivered.

The quality of an online algorithm is measured by the competitive ratio. An online algorithm is called -competitive if, for any input instance, it produces a schedule with an objective value not worse than times the value of an optimal offline schedule.

Restart (see Hoogeveen et al. [1]) means that a running task may be interrupted, losing all the work done on it. The jobs in the interrupted task, which are called restarted jobs, are then released and become independently unscheduled jobs which can be scheduled anew later. Using restarts means that we have a chance to change our original mind to get a better schedule according to the newly arrived jobs and so can efficiently improve the quality of some online algorithms. For example, for the problem of minimizing the time by which all jobs have been delivered on a single machine without restarts, the best possible online algorithm has a competitive ratio of . While using restarts, a best possible online algorithm with a competitive ratio of was given in Akker et al. [2]. Other researches with restarts can see Epstein and Stee [3], Stee and Poutré [4], and Yuan et al. [5].

Limited restart is first introduced by Fu et al. [6]. Limited restart means that once a batch contains some restarted jobs, we cannot restart the batch again. Limited restart implies that a job can be restarted at most once. Limited restart is of practical value. In fact, too many restarts of a job may cause the waste of cost and improve the possibility of a spoiled product.

Parallel batching scheduling is motivated by burn in operations in semiconductor manufacture (see [7, 8]). For online scheduling on a single unbounded parallel batch machine to minimize makespan, that is, the time by which all jobs are completed on the parallel batch machine, Deng et al. [9] and Zhang et al. [10] independently gave the same online algorithm with the competitive ratio of and proved it is best possible. For online scheduling on a single bounded parallel batch machine to minimize makespan, Poon and Yu [11] proved that any FBLPT-based algorithm is -competitive. Particularly for batch capacity , they presented an online algorithm with the competitive ratio of . For online scheduling on unbounded parallel batch machines to minimize makespan, Liu et al. [12] and Tian et al. [13] independently presented two different but best possible online algorithms with a competitive ratio of , where is the positive solution of the equation . For online scheduling on a single parallel batch machine to minimize , that is, the time by which all jobs have been delivered, Tian et al. [14] gave a -competitive online algorithm for the unbounded model and a -competitive online algorithm for the bounded model. Moreover, when each job has the same processing time, they provided two -competitive optimal online algorithms for the two models, respectively. Yuan et al. [15] gave a best possible online algorithm with a competitive ratio of for the two models of and [ implying ]. Fang et al. [16] studied the restricted model that all jobs have their processing times in and provided a class of optimal online algorithms with a competitive ratio of . Tian et al. [17] presented a -competitive online algorithm for the unbounded model. Liu and Lu [18] studied online scheduling on unbounded parallel batch machines to minimize . For the restricted model of [ implying ], they gave a best possible online algorithm with competitive ratio of , where is the positive solution of . For the general case, they gave an online algorithm with competitive ratio of .

For the problem of minimizing makespan on an unbounded parallel batch machine with restarts, Fu et al. [19] showed that any online algorithm cannot have a competitive ratio of less than and Yuan et al. [5] gave a best possible online algorithm matching the lower bound. For the corresponding problem using limited restart, Fu et al. [6] gave an online algorithm with a competitive ratio of and proved that it is best possible. For the problem of minimizing makespan on two unbounded parallel batch machines with limited restart, Fu et al. [20] provided a best possible online algorithm which is -competitive under the second-restart assumption. Recently, Liu and Yuan [21] presented best possible online algorithms for minimizing makespan of equal length jobs on a bounded parallel batch machine with limited restart or restart.

Let be a job set. Then is used to denote the number of jobs in . Let denote the largest delivery time of jobs in and let denote the latest arrival time of jobs whose delivery times are in . For a batch in some schedule, is starting time of on the parallel batch machine. Batch is called full if and nonfull if . In an online algorithm which generates a schedule consisting of processing batches, let denote the th processing batch such that . For simplicity, we write as . Let be the set of the available unscheduled jobs at time in .

If, in an online algorithm , is being processed at the current time , then the following subprocedure which is called Restart may be used.

Restart . At time , interrupt and select jobs in such that each job selected has a delivery time not less than the largest delivery time of jobs not selected in as the batch starting at time . Then reset to be .

This paper studies the online scheduling of equal length jobs on a bounded parallel batch machine with batch capacity to minimize with limited restart. Here means the time by which all jobs have been delivered. This problem can be denoted by . For simplicity, when , the problem is shortly written as P-batch (). When , the problem is shortly written as P-batch ().

Let be the positive solution of and let be the positive solution of .

This paper is organized as follows. In Section 2, we give a best possible online algorithm with a competitive ratio of for problem P-batch (). In Section 3, we give a best possible online algorithm with a competitive ratio of for problem P-batch ().

2. Problem P-Batch

For a job instance , let and denote, respectively, the values of given by an online algorithm and an offline optimal algorithm.

Theorem 1. Any online algorithm for problem P-batch cannot have a competitive ratio of less than .

Proof. Let be a sufficiently small positive number. For any online algorithm , we consider the following instance. At time , the first job with arrives and no jobs arrive before the starting time of .If , no jobs arrive later. We have and . Hence, .If , then, at time , the second job with arrives and no jobs arrive before the starting time of . We distinguish the following two cases.Case 1 (). Then no jobs arrive later. Then and . Thus .Case 2 (). Then interrupts .If is a batch containing only one job , no jobs arrive later. We have and . Hence, .If is a batch consisting of and , then job with arrives at time and no jobs arrive later. As cannot be restarted, we have . In the offline setting, we can schedule starting at time and as a single batch starting at time . So . Then when , we have .If is a batch consisting of and , then job with arrives at time and no jobs arrive later. As cannot be restarted, we have . In the offline setting, we can schedule starting at time and as a single batch starting at time . So . Then when , we have .The theorem follows.

Algorithm . Initially set . At the current time instant , do the following.

: if , , and , then select jobs in such that each job selected has a delivery time not less than the largest delivery time of jobs not selected in as the batch starting at time . Then reset to be .

: if , the machine is idle, and , then select jobs in such that each job selected has a delivery time not less than the largest delivery time of jobs not selected in as the batch starting at time . Then reset to be .

: if the machine is occupied by , , , and each batch starting earlier than is not interrupted, do the following.

: if is nonfull and , then do Restart .

: if , is full, , and some job in has a delivery time larger than the delivery time of some job in , then do Restart .

: otherwise, do nothing but wait.

From algorithm , we have the following observation.

Observation 1. (1) For each batch in algorithm , we have .(2) All jobs arrive at or after . And if , then all jobs arrive at or after .(3) If interrupts , then .(4) If , then .

Theorem 2. For problem P-batch , is a best possible online algorithm with a competitive ratio of .

Proof. Suppose to the contrary that is an instance so that . Let be the schedule generated by running algorithm , for instance, . For simplicity, in the following we always write as and write as without causing confusion. By the definitions of and , we know that is the largest delivery time of jobs in and is the latest arrival time of jobs whose delivery times are in . Let be the earliest starting batch in which assumes and is not interrupted. Then . Let be an offline optimal schedule, for instance, . As , then the assumption that implies .Let ; that is, is the completion time of on the parallel batch machine. Let be the minimum time instant such that, in , there are no idle times between and and each batch processed in the time interval is not interrupted. Suppose that there are batches processed in the time interval in . Then . Set . So .Claim 1 ( does not interrupt and ). Otherwise, interrupts . Then . By algorithm , we have . Let be the job with the largest delivery time among jobs arriving in the interval , let be the delivery time of , and let be the arrival time of . Then . If , then , , , and, accordingly, , a contradiction. If there exists only one batch in the offline optimal schedule , then , and accordingly, , a contradiction. So in the following we assume that , that is, , and assume that there exist at least two batches in the offline optimal schedule . Then and .If , then . By Observation 1, we have . Noticing that , we have , a contradiction.If , then . By algorithm , we know that does not interrupt ; that is, . If , then all jobs in arrive at or after , and so . Thus, , a contradiction. If , then . If, in the offline optimal schedule , is scheduled after the first starting batch, then . Thus, , a contradiction. If, in the offline optimal schedule , is scheduled in the first starting batch and , considering there exist at least two batches in the offline optimal schedule , then . Thus, , a contradiction. If, in the offline optimal schedule , is scheduled in the first starting batch and , then is full and the smallest delivery time of jobs in is not less than . Thus , and so , a contradiction.The above analysis means that does not interrupt . Note that . If , then . From in algorithm , we know that . Thus , a contradiction. Thus, . Claim 1 follows.Claim 2 (some batch in the batches is nonfull). Otherwise, all of the batches are full.If the delivery time of some job in is less than , let be the latest starting batch in such that some job in has a delivery time less than . Then each job in batches starting in the time interval and have delivery times not less than and then arrive after . Then , , and .If or [ and ], then , and so , a contradiction.If and , then we have and . As is full, there exist at least two batches in the offline optimal schedule . Suppose that interrupts , which means that and . If is scheduled in the first starting batch in the offline optimal schedule , then . Thus, , a contradiction. If is scheduled after the first starting batch in the offline optimal schedule , then . Thus, , a contradiction. So we suppose that does not interrupt ; that is, . Note that now . If , then each job in arrives at or later than . Considering that is full, then . Thus, , a contradiction. If , then . If is full, then . Thus, , a contradiction. If is nonfull, considering that does not interrupt , then each job in arrives later than which means that . So , a contradiction.If , noticing that now is full and does not interrupt , then each job in batches starting in the time interval and have delivery times not less than and arrive after by in algorithm . So and . Thus, , a contradiction.So in the following we assume the delivery time of each job in is not less than . Then .If interrupts , then . If , then . Considering and , we have , a contradiction. If , then is nonfull, , and . Note that now there is only one job in . If , considering jobs in the set of all have delivery times not less than , then these jobs arrive after , and, accordingly, . So , a contradiction. If and , then , , is full, and the delivery time of any job in is not less than . And so . So , a contradiction. If and , considering that there is only one job in , then and , and each job in arrives at or after . So . Thus , a contradiction. If and , then and each job in arrives at or after , and so . Thus, , a contradiction.If or [ and does not interrupt ], then there exists some time instant such that and the time interval is an idle time in . Then by , in algorithm , the arrival time of any job in is not less than . Thus , and so , a contradiction. Claim 2 follows.Let be maximum so that is a nonfull batch. From Claim 2, we have .Claim 3 (). Otherwise, suppose that . Since is nonfull, we have , , and . If some batch starting before is interrupted, noticing that is a nonfull batch, then the interrupted batch must start before , and then . Hence, , a contradiction. So in the following we assume that each batch starting before is not interrupted. If , then . As is a nonfull batch and not interrupted, by in algorithm , all jobs in arrive after , which means that . Hence, , a contradiction. If , then , and accordingly, , a contradiction. Claim 3 follows.By the definition of and Claim 3, , are all full and each job in batches starting after arrives after . And .If each job in has a delivery time not less than , then . If , then . Then , a contradiction. If , considering is nonfull and not interrupted, then, by in algorithm , each job in batches starting after arrives after . Thus . Hence, , a contradiction.So in the following we assume that some job in has a delivery time less than . Let be maximum so that some job in has a delivery time less than . Then and batches starting in the time interval are all full. Also each job in batches starting in the time interval and have delivery times not less than and arrive after . Thus . Noting that , we have . If , then . Hence, , a contradiction. If , then , , and . As now is nonfull and not interrupted, then by in algorithm , each job in batches starting after arrives after . Thus . So , a contradiction.From the above discussion we conclude that algorithm has a competitive ratio of at most . By Theorem 1, we know that algorithm is an optimal online algorithm with a competitive ratio of . This completes the proof.

3. Problem P-Batch

Note that is the positive solution of .

Theorem 3. For problem P-batch , there exists no online algorithm with a competitive ratio of less than .

Proof. Theorem 2.4 in Liu and Yuan [21] shows that, for problem with , no online algorithm has a competitive ratio of less than . Here is makespan, namely, the maximum completion time of all jobs on the parallel batch machine. Problem is the special case of . So for problem P-batch (), no online algorithm has a competitive ratio of less than . The theorem follows.

Algorithm . Initially set . At the current time instant , do the following.

: if , , and , then select jobs in such that each job selected has a delivery time not less than the largest delivery time of jobs not selected in as the batch starting at time . Then reset to be .

: if , the machine is idle, and , select jobs in such that each job selected has a delivery time not less than the largest delivery time of jobs not selected in as the batch starting at time . Then reset to be .

: if , , the machine is occupied by , , and each batch starting earlier than is not interrupted, do the following.

: if , is nonfull, , and , then let be the largest delivery time of jobs in . If , or , or [, and ], then do Restart .

: if , is full, , the largest delivery time of jobs in is not less than , and the delivery time of some job in is less than the delivery time of some job in , then do Restart .

: if , is nonfull, , , and {either is full or [ is nonfull and the number of jobs arriving in the time interval is not more than ]}, then do Restart .

: otherwise, do nothing but wait.

From algorithm we have the following observation.

Observation 2. (1) The earliest arrival time of all jobs is not less than . If , then all jobs arrive at or after .(2) If or , is nonfull, and the number of jobs arriving in the time interval is more than , then there exists no batch which is interrupted.(3) If , then .(4) If , then .

Theorem 4. For problem P-batch , is a best possible online algorithm with a competitive ratio of .

Proof. Suppose to the contrary that is an instance so that . Let be the schedule generated by running algorithm for the instance and let be an offline optimal schedule for the instance . As the proof in Theorem 2, is used to denote the largest delivery time of jobs in and is the latest arrival time of jobs whose delivery times are in . Let be the earliest starting batch in which is not interrupted and assumes . Then . As , then the assumption that implies .Let ; that is, is the completion time of on the parallel batch machine. Let be the minimum time instant such that, in , there are no idle times between and and each batch processed in the time interval is not interrupted. Suppose that there are batches processed in the time interval in . Then . Set . So .Claim 1 ( does not interrupt and ). Otherwise, is a restricted batch. By algorithm , we have and ; that is, . Let be the largest delivery time of jobs arriving in the interval , let be the job with delivery time , and let be the arrival time of . Then . If , then , , , and, accordingly, , a contradiction. So in the following we assume that ; that is, . Then . If, in the offline optimal schedule , there exists only one batch, or and are scheduled in one batch, or the batch containing starts later than the batch containing , then . Considering , then , a contradiction. So in the following we assume that, in the offline optimal schedule , there exist at least batches, and are scheduled in two different batches, and the batch containing starts earlier than the batch containing . Then we easily get .If and is full, then , , and . By Observation 2, we have . Considering , then , a contradiction.If and is nonfull, then . If , then . Noticing that , then , a contradiction. Hence we have . As interrupts , considering and , then, by in algorithm , the only possibility is that . Then . Thus, , a contradiction.If , then and . Also noticing that now does not interrupt , we have . Note that and in the offline optimal schedule , the batch containing starts earlier than the batch containing . If , then , , is full and each job in has a delivery time not less that , and, accordingly, . Thus, , a contradiction. If , then . Thus , a contradiction. If , then , , and . Thus , a contradiction. So in the following we assume that . Then , , and . If , then , a contradiction. If , noticing that does not interrupt and interrupts , by in algorithm , then is full. If each job in the full batch has a delivery time not less than , then , and then , a contradiction. So some job in the full batch has a delivery time less than . As is full, , some job in has a delivery time less than , and does not interrupt , then , and so . Thus, , a contradiction.The above analysis means that does not interrupt . Note that . If , then . From in algorithm , we know that . Thus , a contradiction. Hence, . Claim 1 follows.Claim 2 (some batch in the batches is nonfull). Otherwise, all of the batches are full.If the smallest delivery time of jobs in is not less than and interrupts , then . If and is full, then and . Thus, considering and , we have , a contradiction. If [ and is nonfull] or , then is nonfull and each job in batches starting after arrives after . Then and . Thus , a contradiction.If the smallest delivery time of jobs in is not less than and or [ and does not interrupt ]}, then the arrival time of any job in is not less than . Thus , and so , a contradiction.So in the following we suppose that the delivery time of some job in is less than . Let be the latest starting batch in such that some job in has a delivery time less than . Then each job in batches starting in the time interval and have delivery times not less than and, accordingly, arrive after . Then , , and .If or [ and ], then , and, accordingly, , a contradiction.If , , and is nonfull or [ is full and does not interrupt ]}, then . Thus, , a contradiction. If , , is full, and interrupts , then some job with a delivery time not less than arrives after and . Thus, , a contradiction.If and , then , and, accordingly, , a contradiction. If and , then there are at least two batches in the offline optimal schedule . Note that now . If is scheduled in the first batch in the offline optimal schedule , then . Thus, , a contradiction. If is scheduled in the batch starting after the first batch in the offline optimal schedule , then . Thus, , a contradiction. Claim 2 follows.Let be maximum so that is a nonfull batch. From Claim 2, we have .Claim 3 (). Otherwise, suppose that . As is a nonfull batch, then , , and .If , noticing that is a nonfull batch, then . Then , a contradiction.If , then , , , and . If , then , a contradiction. So we suppose that . If , then . Thus , a contradiction. So in the following we assume that . If there is only one batch in the offline optimal schedule , then and . Also considering that does not interrupt , we have that and . Then . And , a contradiction. So in the following we assume that there are at least two batches in the offline optimal schedule . If is scheduled in the first starting batch in the offline optimal schedule , then . Thus, . If is scheduled in the batch starting after the first starting batch in the offline optimal schedule , then . Thus, , a contradiction. Claim 3 follows.By the definition of and Claim 3, , are all full and each job in batches starting after arrives after . And .If each job in has a delivery time not less than , then . Then , a contradiction. So in the following we assume that some job in has a delivery time less than . Let be maximum so that some job in has a delivery time less than . Then and batches starting in the time interval are all full. And each job in batches starting in the time interval and have delivery times not less than and then arrive after . Thus . Note that . Considering , then . Hence, , a contradiction.From the above discussion we conclude that algorithm has a competitive ratio of at most . By Theorem 3 we know that algorithm is an optimal online algorithm with a competitive ratio of . This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Sciences Foundation (Grant no. 20142BAB211017) of Jiangxi Province and the School Subject (Grant no. 06162015) of Jiangxi University of Finance and Economics.